Note: As it stands this is a mathematical question, and not really about physics. I want to know about the setting up of a certain integral and showing that it equals $p dV$.
As is well known, the First Law of Thermodynamics is given by $$dU = dQ - p dV$$
Here $dU$ is an exact differential form, whereas both $dQ$ and $p dV$ aren't. In elementary texts, the infinitesimal work done by a gas enclosed in a cylindrical piston is calculated as $dW = p dV$. In only two books (one by Fermi, and another by Sommerfeld) are sketches of proof given for a gas enclosed in a container with any shape whatsoever.
The sketches given are not exactly mathematically rigorous. There are suggestions that a surface integral should be computed, but I can't seem to formulate it in terms of vector calculus or differential forms.
In fact this question has been raised before (and the corresponding passage in Fermi included, see here : How can this result in Thermodynamics be rigorously proved?), but so far there doesn't seem to be any satisfactory answer.
I would like to know how this is done in a mathematically precise way.